Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 703539, 4 pages
http://dx.doi.org/10.1155/2014/703539
Research Article

New Periodic Solutions for the Singular Hamiltonian System

School of Business Administration, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 30 May 2014; Accepted 20 October 2014; Published 9 November 2014

Academic Editor: Jianqing Chen

Copyright © 2014 Yi Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.

1. Introduction

We are mainly interested in the existence of periodic solutions , with a prescribed period, of the second-order differential equation with (, ) and where denotes the gradient of the function defined on . The study of the periodic solutions of such equations has a substantial literature with the works [121] of particular importance for our purpose.

In the 1975 paper of Gordon [10], variational methods were used to study periodic solutions of planar 2-body type problems under assumptions which have come to be known as Gordon’s Strong Force condition ():there exists a neighborhood of 0 and a function such that(i);(ii) for every and ;(iii).

For 2-body type problems in , one can see the works of Ambrosetti-Coti Zelati, Bahri-Rabinowitz, Greco, and other mathematicians from [15, 1821]. In this paper we wish to highlight two main results among them, that is, Theorems 1 and 2. Firstly, we must specify three separate conditions may satisfy about its behavior at infinity. Suppose that is -periodic in ; then, (uniformly for ) and for every , ;there exist such that, for every and with ,(i);(ii);there exist , , , such that, for every , ,(i);(ii).

Set for every ; there hold the following results.

Theorem 1 (Greco [11]). If and and one of hold, then in (1) there is at least one nonconstant -periodic solution.

Theorem 2 (Bahri-Rabinowitz [3] and Greco [11]). Suppose , so that , and the following condition holds:
is compact (or empty).
If and one of hold, then in (1) there exist infinitely many nonconstant -periodic solutions.

In this paper, we prove the following new theorem.

Theorem 3. Suppose satisfies condition and conditionsfor the given , ;there exists such that, , as uniformly for .

Then the system (1) has a -periodic solution.

Corollary 4. Suppose , , , , , and then, , the system (1) has a -periodic solution. From the above example, we see that our potential does not satisfy any of conditions , , and .

2. A Few Lemmas

In order to prove Theorem 3, we will need to recall the following useful lemmas.

Lemma 5 (Sobolev-Rellich-Kondrachov [15]). It is well known that and the imbedding is compact.

Lemma 6 (Eberlein-Shmulyan [15]). A Banach space is reflexive if and only if every bounded sequence in has a weakly convergent subsequence.

Lemma 7 (Ekeland [8]). Let be a Banach space, and suppose defined on is Gateaux-differentiable, lower semicontinuous, and bounded from below. Then there is a sequence such that

Definition 8 (see [8]). Let be a Banach space and . We say satisfies the condition if whenever such that then has a strongly convergent subsequence.

Interestingly, Cerami [22] considers a weaker compact condition on a Banach space than the classical condition. Here we introduce a similar condition in an open subset of a Banach space.

Definition 9 (see [8]). Let be a Banach space; is an open subset; and suppose defined on is Gateaux-differentiable. We say that satisfies the condition if whenever is a sequence such that then has a strongly convergent subsequence in .

With this definition, we can deduce a minimizing result in an open subset of a Banach space, the proof of which is similar to the standard one.

Lemma 10 (see Mawhin-Willem [15]). Let be a Banach space, an open subset, and . Assume has a lower bound on the closure of , and let . If satisfies on and as , then is a critical value for .

Lemma 11 (see [10]). Let , and satisfies the Gordon’s Strong Force condition ; let Define Then as .

3. The Proof of Theorem 3

Let and .

Lemma 12 (see [2]). Suppose satisfies condition and define Then the critical point of is a -periodic solution of (1).

Lemma 13. If satisfies and in Theorem 1, then satisfies the Cerami-Palais-Smale condition for any ; that is, for any , if then has a strongly convergent subsequence and the limit is in .

Proof. By condition and Lemma 11, we must have as . Since , we know that, for any given , there exists such that when , there holds the inequality The limit implies and so Using condition together with the limits and inequalities (11), (12), and (14), we can choose such that when is large enough, there holds which implies is bounded.
In the following we prove that is bounded; otherwise, there is a subsequence, still denoted by , such that Then by Newton-Leibniz’s formula, we have Now, by and , we have which contradicts the limit (14).
Hence, is bounded; has a weakly convergent subsequence; we still denote it by , and let the limit be . We can show in a standard fashion that this subsequence is strongly convergent in . To complete the proof, we write it out.
Since the sequence is bounded in , so, by Sobolev’s embedding inequality, we know it is also bounded in maximum norm, and, by condition and Lemma 11, we know that when is large, By , when is large, is also uniformly bounded in maximum norm; we have Taking and in the above equation, we get Since , hence ; furthermore, since is bounded, so . Hence, by (21) and the uniformly bounded property for , we have By weakly, we have By Sobolev Embedding Theorem, has a subsequence, still denoted by subject to .
We notice That is, strongly in . Then by Lemma 10 the proof of Theorem 3 is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author sincerely thanks the referee for his/her valuable comments and suggestions.

References

  1. A. Ambrosetti and V. C. Zelati, “Closed orbits of fixed energy for singular Hamiltonian systems,” Archive for Rational Mechanics and Analysis, vol. 112, no. 4, pp. 339–362, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Springer, 1993.
  3. A. Bahri and P. H. Rabinowitz, “A minimax method for a class of Hamiltonian systems with singular potentials,” Journal of Functional Analysis, vol. 82, no. 2, pp. 412–428, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. V. Benci and F. Giannoni, “Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials,” Journal of Differential Equations, vol. 82, no. 1, pp. 60–70, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. C. Carminati, E. Séré, and K. Tanaka, “The fixed energy problem for a class of nonconvex singular Hamiltonian systems,” Journal of Differential Equations, vol. 230, no. 1, pp. 362–377, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Degiovanni and F. Giannoni, “Dynamical systems with Newtonian type potentials,” Annali della Scuola Normale Superiore di Pisa, vol. 15, no. 3, pp. 467–494, 1988. View at Google Scholar · View at MathSciNet
  8. I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, New York, NY, USA, 1990. View at MathSciNet
  9. E. Fadell and S. Husseini, “A note on the category of the free loop space,” Proceedings of the American Mathematical Society, vol. 107, no. 2, pp. 527–536, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. B. Gordon, “Conservative dynamical systems involving strong forces,” Transactions of the American Mathematical Society, vol. 204, pp. 113–135, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  11. C. Greco, “Periodic solutions of a class of singular Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 12, no. 3, pp. 259–269, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. E. W. van Groesen, “Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,” Journal of Mathematical Analysis and Applications, vol. 132, no. 1, pp. 1–12, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 2nd edition, 1952. View at MathSciNet
  14. P. Majer, “Ljusternik-Schnirelmann theory with local Palais-Smale conditions and singular dynamical systems,” Annales de l'Institut Henri Poincaré: Analyse Non Linéaire, vol. 8, no. 5, pp. 459–476, 1991. View at Google Scholar · View at MathSciNet
  15. J. Mawhin and M. Willem, Critical Point Theory and Applications, Springer, 1989.
  16. L. Pisani, “Periodic solutions with prescribed energy for singular conservative systems involving strong forces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 21, no. 3, pp. 167–179, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. P. H. Rabinowitz, “A note on periodic solutions of prescribed energy for singular Hamiltonian systems,” Journal of Computational and Applied Mathematics, vol. 52, no. 1–3, pp. 147–154, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. K. Tanaka, “A prescribed energy problem for a singular Hamiltonian system with a weak force,” Journal of Functional Analysis, vol. 113, no. 2, pp. 351–390, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. K. Tanaka, “A prescribed-energy problem for a conservative singular Hamiltonian system,” Archive for Rational Mechanics and Analysis, vol. 128, no. 2, pp. 127–164, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. K. Tanaka, “Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds,” Annales de l'Institut Henri Poincaré, vol. 17, no. 1, pp. 1–33, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. S. Terracini, “Multiplicity of periodic solution with prescribed energy to singular dynamical systems,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 9, no. 6, pp. 597–641, 1992. View at Google Scholar · View at MathSciNet
  22. G. Cerami, “Un criterio di esistenza per i punti critici so variete illimitate,” Istituto Lombardo, Accademia di Scienze e Lettere, Rendiconti. Scienze Matematiche e Applicazioni A, vol. 112, pp. 332–336, 1978. View at Google Scholar